At what point or points in its path does a projectile have its minimum speed? Its maximum?

In projectile motion, the horizontal component of velocity always remains the same and will contribute to the net velocity of the object in motion. The presence of vertical component will add to the magnitude of the net velocity of the object and therefore the object will have minimum velocity at those points in which the vertical component of velocity gets to zero. This happens when the object reaches its maximum attainable height.
Therefore, the minimum velocity of the object in projectile motion will be at the highest point.
The magnitude of the vertical component of velocity varies with time such that it is maximum when the object is released, then it decreases as the object reaches the maximum height, and finally increases again to hit the ground with the same velocity with which it was released.
Therefore the vertical component of velocity is maximum at two points: when the object was released initially and when the object is about to hit the ground. The horizontal component of velocity remains constant throughout. This ensures that the magnitude of net velocity is maximum at those points where the magnitude of vertical component of velocity of the object is maximum.
Hence, the object has maximum velocity at the beginning of the projectile motion, when it was thrown intially and when it is about to hit the ground, at the end of his motion.